Question: Solve for $x$ and $y$ by deriving an expression for $y$ from the second equation, and substituting it back into the first equation. $\begin{align*}-2x-6y &= -2 \\ x-9y &= -6\end{align*}$
Solution: Begin by moving the $x$ -term in the second equation to the right side of the equation. $-9y = -x-6$ Divide both sides by $-9$ to isolate $y$ $y = {\dfrac{1}{9}x + \dfrac{2}{3}}$ Substitute this expression for $y$ in the first equation. $-2x-6({\dfrac{1}{9}x + \dfrac{2}{3}}) = -2$ $-2x - \dfrac{2}{3}x - 4 = -2$ Simplify by combining terms, then solve for $x$ $-\dfrac{8}{3}x - 4 = -2$ $-\dfrac{8}{3}x = 2$ $x = -\dfrac{3}{4}$ Substitute $-\dfrac{3}{4}$ for $x$ back into the top equation. $-2( -\dfrac{3}{4})-6y = -2$ $\dfrac{3}{2}-6y = -2$ $-6y = -\dfrac{7}{2}$ $y = \dfrac{7}{12}$ The solution is $\enspace x = -\dfrac{3}{4}, \enspace y = \dfrac{7}{12}$.